Problem 71 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } w: A=l \cdot w $$

Step-by-Step Solution

Answer

The solution is \( w = \frac{A}{l} \).

Step 1: Understand the given equation

We are given the equation \( A = l \cdot w \), where \( A \) is the area, \( l \) is the length, and \( w \) is the width. We need to solve this equation for \( w \).

Step 2: Isolate the variable w

To solve for \( w \), we need to isolate \( w \) on one side of the equation. We do this by dividing both sides of the equation by \( l \).

Step 3: Rewrite the equation

Divide both sides of the equation \( A = l \cdot w \) by \( l \): \[ w = \frac{A}{l} \]

Step 4: Interpret the solution

The equation \( w = \frac{A}{l} \) tells us that the width \( w \) is equal to the area \( A \) divided by the length \( l \). This solution finds \( w \) in terms of \( A \) and \( l \).

Key Concepts

Solving for a Variable

When you're working with algebraic equations, one common task is solving for a specific variable. In the given exercise, we're asked to solve for the variable \( w \) in the equation \( A = l \cdot w \). This means we need to express \( w \) in terms of the other variables given in the equation.To solve for a variable means to manipulate the equation so that the variable of interest is isolated on one side. The solution will show how this variable relates to other quantities in the equation. This is useful in applied scenarios, like determining dimensions from area measurements, which is exactly what we're doing here by finding \( w \) in terms of \( A \) and \( l \). It often involves using basic algebraic operations such as addition, subtraction, multiplication, or division to rearrange the equation. Unlocking a variable in an equation allows you to effectively solve real-world problems by substituting in known values of the other variables. This helps build a bridge between abstract mathematical relationships and practical applications.

Area Formula

The area formula \( A = l \cdot w \) is foundational in geometry and helps calculate the space contained within a two-dimensional shape, such as a rectangle. Here, \( A \) represents the area, \( l \) is the length, and \( w \) is the width.Understanding this formula is crucial when you are given certain dimensions and need to find another. The equation shows that area is the product of length and width. Knowing this can help you solve a variety of problems related to real-life situations, such as when measuring materials for construction or when determining the size of a garden plot.Moreover, solving for a different variable in this formula, such as \( w \) in our example, helps demonstrate the flexibility and power of algebra in rearranging formulas to find the missing information you need. It's a vital skill that emphasizes interdependency of the variables, showing how changes in one dimension affect the entire shape.

Isolation of Variables

Isolation of variables refers to the process of rearranging an equation to display a specific variable by itself on one side of the equation. In the exercise, we applied this to the equation \( A = l \cdot w \) to isolate \( w \). To do this, you divide both sides of the equation by \( l \), resulting in \( w = \frac{A}{l} \). This is an essential skill in algebra since it enables you to transform equations and find solutions that express one variable in terms of others. The primary goal is to systematically "undo" the operations surrounding the variable you're solving for. In this case, multiplication by \( l \) was undone by division.Here are some tips to remember:

• Identify the variable you need to isolate.
• Perform inverse operations to move other terms to the opposite side of the equation.
• Simplify the equation step by step until the desired variable stands alone.

Mastering the isolation of variables not only helps in solving textbooks problems but also equips you with the tools to handle complex equations in various scientific and engineering fields.